Past Events

• Francesca Arici (Universiteit Leiden): SU(2)-symmetries of C*-algebras, subproduct systems, and K-theory

This talk is motivated by multivariate operator theory and by the study of symmetries in the context of C*-algebras. We shall consider SU(2)-equivariant subproduct system of Hilbert spaces and their Toeplitz and Cuntz–Pimsner algebras. We will provide results about their topological invariants through K(K)- theory. More specifically, we will show that the Toeplitz algebra of the subproduct system of an SU(2)-representation is equivariantly KK-equivalent to the algebra of complex numbers so that the (K)K- theory groups of the Cuntz–Pimsner algebra can be effectively computed using a Gysin exact sequence involving an analogue of the Euler class. Finally, we will discuss why and how C*-algebras in this class satisfy Poincaré duality.

Based on joint work with Jens Kaad (SDU Ondense), Yufan Ge (Leiden), and Dimitris Gerontogiannis (Leiden).

• Christian Bönicke (Newcastle University): Homology and K-theory of totally disconnected dynamical systems

Groupoids form a convenient framework to study various kinds of (topological) dynamical systems. They also play an increasingly important role in the abstract theory of C*-algebras and their classification.  This talk will focus on homological and K-theoretical invariants of groupoids. In particular, I will give an overview on some recent progress on a conjecture posed by Matui that relates the K-theory of a groupoid C*-algebra to the homology of the underlying groupoid itself.

• Samantha Pilgrim (University of Glasgow): Crossed Product C*-algebras and Finite Approximations of Topological Dynamical Systems

We will discuss how finite dimensional approximations of crossed product C^*-algebras can arise from approximations of topological dynamical systems.  In particular, we will give an exposition of residual finiteness and quasidiagonality of group actions in the sense of Kerr and Nowak; and present theorems on finite approximations of isometric actions.  As corollaries, we obtain economical proofs of quasidiagonality for crossed products of isometric actions by amenable groups as well as many new examples of crossed products by non-amenable groups with the MF property.  Interactions with semiprojectivity properties of C*-algebras may also be discussed, time permitting.  

• Ian Charlesworth (Cardiff University): Regularity in Free Probability and Free Stein Dimension

The study of regularity in free probability boils down to the question of how much information about a *-algebra can be gleaned from probabilistic properties of its generators. Some of the first results in this theme come from the theory of Voiculescu's free entropy: generators satisfying certain entropic assumptions generate von Neumann algebras which are non-Gamma, or prime, or do not admit Cartan subalgebras. Free Stein dimension -- a quantity I introduced with Nelson -- is a more recent quantity in a similar vein, which is robust under polynomial transformations and not trivial for variables which do not embeddable in R^\omega.

After giving a brief introduction to free probability and free entropy, I will speak on some recent improvements in related to free Stein dimension. We are now able to compute the free Stein dimension of direct sums, and of tensor products with finite dimensional algebras, which allow us to compute it in a large number of new examples. We are also able to give bounds on Stein dimension based on the presence of algebraic relations among generators. This is joint work with Brent Nelson.

• Kevin Boucher (University of Southampton): Uniformly bounded representations of hyperbolic groups

After a brief introduction to subject of spherical representations of hyperbolic groups, I will present a new construction motivated by a spectral formulation of the so-called Shalom conjecture.

This is joint work with Jan Spakula.

• Cornelia Drutu (Oxford): Actions of groups on Banach spaces
  Abstract: This talk will discuss various versions of fixed point properties, generalizing property (T), and of proper actions on Banach spaces, generalizing a-T-menability. In particular, I will describe a notion of spectrum providing an optimal way to measure ``the strength'' of the property (T) that an infinite group may have, and what can be said about this spectrum, in particular for hyperbolic groups. I will also describe weak versions of a-T-menability for (acylindrically) hyperbolic groups and for mapping class groups. This is on joint work with Ashot Minasyan and Mikael de la Salle, and with John Mackay.

• Steven Flynn (Bath): Inhomogeneous Pseudodifferential Calculus with noncommutative symbols
  Abstract: I will summarize various developments of inhomogeneous pseudodifferential calcului adapted to the study of hypoelliptic operators. This includes the constructions by Van Erp/Yuncken and Choi/Ponge of the tangent groupoid for filtered manifolds. I will then explain how operator-valued symbols arise naturally, and how such symbols are used to characterize the asymptotic behavior of high energy solutions to the Schrödinger equation with a hypoelliptic Hamiltonian. 

• Maryan Hosseini (QMUL): Dimension Groups and Topological Factoring of Bratteli-Vershik Systems
  Abstract: Bratteli diagrams were initially introduced in classifications of Af-algebras. In 1992, R. H. Herman I. Putnam and C. Skau had shown the close connections between Cantor minimal dynamical systems, Bratteli diagrams and Dimension groups which led to many studies on Cantor dynamics. Knowing topological factors of a dynamical systems is one of the interests of people in the field. Topological factors of a dynamical system can realize topological or sometimes all measurable spectrum of the system and may give information about the complexity of the system. In 2019, M. Amini, G. Elliott and N. Golestani proved that existence of topological factoring between two Cantor minimal systems is equivalent to the existence of some morphism between their Bratteli-Vershik realizations. In this talk, I will talk about some applications of this equivalence in the study of Cantor minimal systems and by time permission will discuss whether existence of such morphism will make implications toward existence of factoring between two systems from a larger family of Cantor systems. The talk is mostly based on my previous and a current joint work with N. Golestani. 

• Shanshan Hua (Oxford): Nonstable K-theory for Z-stable C*-algebras
  Abstract: In Jiang's unpublished paper (1997), it is shown that any Z-stable C*-algebra A is K1-injective and K1-surjective, which means that K1(A) can be calculated by looking at homotopy equivalence classes of U(A), without matrix amplifications. Moreover, for such A, higher homotopy groups of U(A) are isomorphic to K0(A) or K1(A), depending on the dimension of the higher homotopy group. In this talk, I will present Jiang's result for Z-stable C*-algebras. Moreover, I will explain briefly our new strategies to reprove his theorems using an alternative picture of the Jiang-Su algebra as an inductive limit of generalized dimension drop algebras.

• Adrian Ioana (UC San Diego): Wreath-like product groups and rigidity of their von Neumann algebras
  Abstract: Wreath-like products are a new class of groups, which are close relatives of the classical wreath products. Examples of wreath-like product groups arise from every non-elementary hyperbolic groups by taking suitable quotients. As a consequence, unlike classical wreath products, many wreath-like products have Kazhdan's property (T).
  In this talk, I will present several rigidity results for von Neumann algebras of wreath-like product groups. We show that any group G in a natural family of wreath-like products with property (T) is W*-superrigid: the group von Neumann algebra L(G) remembers the isomorphism class of G. This provides the first examples of W*-superrigid groups with property (T). For a wider class wreath-like products with property (T), we show that any isomorphism of their group von Neumann algebras arises from an isomorphism of the groups.  As an application, we prove that any countable group can be realized as the outer automorphism group of L(G), for an icc property (T) group G. These results were obtained in joint works with Ionut Chifan, Denis Osin and Bin Sun.
  I will also mention an additional application of wreath-like products obtained in joint work with Ionut Chifan and Daniel Drimbe, and showing that any separable II_1 factor is contained in one with property (T). This provides an operator algebraic counterpart of the group theoretic fact that every countable group is contained in one with property (T).

• Sam Richardson (Cardiff): Natural Transformations of Cohomology Theories Induced by Exponential Functors
  Abstract: Exponential functors are monoidal functors that transform the direct sum of vector spaces into the tensor product of two slightly different vector spaces. The source and target categories of an exponential functor are both strict symmetric monoidal categories and so we may use them to construct cohomology theories.
  Of particular note for C*-algebras is the map induced in the 1st degree. We take the geometric realisation of the nerve of our exponential functor to achieve a map of spaces and it just so happens that the target space is homotopy equivalent to the classifying space of the automorphism group of a certain C*-algebra also derived from the exponential functor.
  To better understand the exotic cohomology theories we achieve we investigate the class of the Weyl map W: SU(n)/T x T -> SU(n) which will prove very useful thanks to it’s particular product structure and the suspension-loop adjunction. Since the map we are studying factors through the space U we will be able to investigate using K theory first and then hopefully it will be as easy as applying our exponential functor to the relevant vector bundles.